To solve part (a) and determine the moment of inertia $I$ of the system (child and carousel), let us proceed step by step:

Given:

1. Moment of inertia of the carousel: $I_{\text{carousel}} = 2mR^2$, where $m$ is the mass of the carousel and $R$ is its radius.
2. The child of mass $m$ is sitting at the edge of the carousel at radius $R$.

The moment of inertia of the entire system is the sum of the moments of inertia of the carousel and the child:

$I_{\text{total}} = I_{\text{carousel}} + I_{\text{child}}.$

Moment of inertia of the child:

The child is treated as a point mass located at a distance $R$ from the center. The moment of inertia for a point mass is given by:

$I_{\text{child}} = m \cdot R^2.$

Moment of inertia of the entire system:

$I_{\text{total}} = I_{\text{carousel}} + I_{\text{child}} = 2mR^2 + mR^2 = 3mR^2.$

Thus, the moment of inertia of the system is:

$\boxed{I_{\text{total}} = 3mR^2}.$

Let me know if you want explanations or clarifications for part (b) or (c).

Follow-up questions:

1. How does the moment of inertia change if the child moves closer to the center?
2. How would the angular velocity $\omega$ of the system be affected by the child’s position?
3. What happens to the kinetic energy of the system if the angular velocity remains constant, but the child changes position?
4. How would the moment of inertia change if the carousel's mass is doubled?
5. What is the relationship between torque and angular acceleration for this system?

Tip:

Always remember that for composite systems, the total moment of inertia is the sum of individual contributions, based on their distances from the axis of rotation.